Qualitative Korovkin-type results on almost convergence

Qualitative Korovkin-type results on almost convergence

Juan Hern´ andez Guerra

and Daniel C´ ardenas-Morales

1 Depto. de M´etodos Cuantitativos. Univ. de Las Palmas de Gran Canaria.

2 Departamento de Matem´aticas. Universidad de Ja´en.

[email protected], [email protected]

Monograf´ıas del Semin. Matem. Garc´ıa de Galdeano. 27: 331–336, (2003).

Abstract

In this paper we state a general Korovkin-type result for the study of the al- most convergence of general shape preserving approximation processes. We use the well-known notion of almost convergence introduced by Lorentz in 1948. Some applications are also shown.

Keywords: Korovkin-type result, almost convergence AMS Classification: 41A35, 41A36

1 Introduction

The classical results of Korovkin [3] and the subsequent quantitative versions of Shisha and Mond [7], [8] on the convergence of a sequence of positive linear operators were formulated in [2] and [5] replacing the usual convergence by almost convergence. We recall here this notion that was introduced in 1948 by Lorentz [4]. As usual we denote by R^X the set of all real-valued functions on a set X, and by C(X) the subset of the continuous ones.

Let H_n be a sequence of linear operators defined on C(X). Given f ∈ C(X), we define Hnf := Hn(f ) to be almost convergent to g in C(X), uniformly in X, (Hnf −→ g),^a.c.

provided

1 p

v+p

X

n=v+1

H_nf (x), p, v ∈ N

converges to g(x) as p → ∞, uniformly in v and uniformly in X. Pointwise almost convergence can be defined in an obvious way.

On the other hand, in [6] it was stated a Korovkin-type result on the convergence of sequences of operators that possess shape preserving properties much more general than positivity.

Our aim with this paper is to present qualitative results on almost convergence for these conservative approximation processes that we have just mentioned and that we detail

in the next section together with the main result of this work. The rest of the sections are devoted to present applications of this result to the space of k-times differentiable functions defined on a compact subinterval of R and to the space C2π of all real-valued continuous 2π periodic functions on R.

2 A general Korovkin-type result on almost conver- gence

In [6] it was presented an extension of the classical Korovkin theorem when non positive operators are considered. In this section we extend this theorem for the case of almost convergence.

Let m ∈ N, let X be a compact subset of R^m, let B ⊂ R^X, let A be a subspace of C(X) such that A ⊂ B, and let L : B → R^X be a linear operator such that L(A) ⊂ C(X).

Theorem 1 Let P = {f ∈ B : Lf ≥ 0} and let C be a cone of A (i.e. if α ≥ 0 and f, g ∈ C, then αf, f + g ∈ C). Assume that:

(v.1) there exists u ∈ A such that Lu(x) = 1 ∀x ∈ X.

(v.2) there exist ai, gi ∈ C(X), i = 1, 2, ..., m, such that for all z ∈ X, the functions ϕ_z :=

m

X

i=1

a_i(z)g_i

belong to C, Lϕ_z(x) ≥ 0 for all x ∈ X and the equality is satisfied if and only if z = x.

(v.3) for each f ∈ A, there exists α = α(f ) ≥ 0 such that if β > α, then f + βϕ_z ∈ C.

Let Kn : A → B be a sequence of linear operators such that (k.1) K_n(P ∩ C) ⊂ P ,

(k.2) L(K_ng_i)−→ Lg^a.c. _i, i = 1, . . . , m.

Then for all f ∈ A,

L(K_nf )−→ Lf.^a.c.

Proof We organize the proof by several steps in a similar way as it is done in [1, Theorem 1.3]. For g ∈ A, p, v ∈ N, we use the notation t^vpg(x) := ¹_p^Pv+p_n=v+1K_ng(x).

1. It is verified that L(K_nϕ_z)(z)−→ 0, ∀z ∈ X.^a.c.

Indeed, applying the definition of almost convergence, L(K_nϕ_z)(z) −→ 0 ⇐⇒ L(t^{a.c. v}_pϕ_z)(z) = 1

p

v+p

X

n=v+1

L(K_nϕ_z)(z) → 0,

as p → ∞, uniformly in v, so it suffices to observe that 1

p

v+p

X

n=v+1

L(K_nϕ_z)(z) = 1 p

v+p

X

n=v+1 m

X

i=1

a_i(z)L(K_ng_i)(z)

!

=

m

X

i=1

a_i(z)





1 p

v+p

X

n=v+1

L(K_ng_i)(z)



−→

m

X

i=1

a_i(z)Lg_i(z) = ϕ_z(z) = 0, where we have used hypotheses (v.2) and (k.2).

2. L(t^v_pu) is bounded on X.

In order to prove this assertion, let z₁, z₂ ∈ X (z₁ 6= z₂) and define ϕ^∗ = ϕ_z1+ ϕ_z2. ¿From (v.2), it is obvious that Lϕ^∗(x) > 0, ∀x ∈ X. As X is a Haussdorf space, then there exist δ₁, δ₂ > 0, such that B(z₁, δ₁) ∩ B(z₂, δ₂) = ∅. Let µ₁, µ₂ be the minimum values of ϕ_z1 y ϕ_z2 in X − B(z₁, δ₁) and X − B(z₂, δ₂) respectively, both necessarily positive. Defining µ = min{¹₂µ₁,¹₂µ₂}, one has that for all x ∈ X

Lu(x) = 1 ≤ µ⁻¹Lϕ^∗(x).

Hence µ⁻¹ϕ^∗ − u ∈ P . Moreover, according to hypothesis (v.3), for a sufficiently small value of β, with β ≤ µ, we obtain β⁻¹ϕ^∗−u ∈ C. Thus, β⁻¹ϕ^∗−u ∈ P ∩C. Applying (k.1), it is obtained that L(K_n(β⁻¹ϕ^∗− u)) ≥ 0 and directly β⁻¹L(t^v_pϕ^∗) ≥ L(t^v_pu). Using (k.2), as p → ∞, β⁻¹L(t^v_pϕ^∗) = β⁻¹L(t^v_pϕ_z1) + β⁻¹L(t^v_pϕ_z2) −→ β⁻¹Lϕ_z1 + β⁻¹Lϕ_z2 = β⁻¹Lϕ^∗, uniformly in v. Thus, L(t^v_pu) is bounded by a convergent and continuous function defined on a compact set. Therefore, L(t^v_pu) is bounded.

3. Let f_y ∈ A, y ∈ X, be a family of functions for which Lf_y(x) is a continuous function of (x, y) ∈ X × X and Lf_y(y) = 0, ∀y ∈ X. Then L(K_nf_y)(y) −→ 0 uniformly^a.c.

in y ∈ X.

This assertion is proved as follows: let D = {(y, y)/y ∈ X} in X × X and let y ∈ X, then it is verified that for all  > 0 there exists a neighborhood Vy of the point (y, y) such that

|Lf_y(x)| < , ∀(x, y) ∈ V_y. Let V = ∪_y∈XV_y and let F its complement in X × X which is clearly compact. Thus we can consider

m = min_(x,y)∈FL(ϕ_y)(x), M = max_(x,y)∈FLf_y(x),

and write |Lf_y(x)| <  + ^M_mLϕ_y(x) for all x, y ∈ X. From (v.3), for a sufficiently large β, it is verified that u + β^M_mϕ_y ± f_y ∈ P ∩ C. Applying the hypothesis (k.1), we obtain

K_nu + βM

mK_nϕ_y ± K_nf_y ∈ P . Thus,

−L(K_nu)(y) − βM

mL(K_nϕ_y)(y) < L(K_nf_y)(y) < L(K_nu)(y) + βM

mL(K_nϕ_y)(y), and directly

−L(t^v_pu)(y) − βM

mL(t^v_pϕ_y)(y) < L(t^v_pf_y)(y) < L(t^v_pu)(y) + βM

mL(t^v_pϕ_y)(y).

Therefore

L(t^v_pf_y)(y)< L(t^v_pu)(y) + βM

mL(t^v_pϕ_y)(y).

¿From the steps 1 and 2 of this proof, there exists U₀ ∈ R, upper bound of L(t^vpu), and p₀ ∈ N, such that for all p > p0

L(t^v_pf_y)(y)< U₀+  = (U₀+ 1), for all v ∈ N and for all y ∈ X.

4. Now we are in a position to prove the theorem. Given f ∈ A, we define the family fy = f − Lf (y)

Lϕ^∗(y)ϕ^∗, ∀y ∈ X.

The functions Lf_y(x) are continuous in x and y with Lf_y(y) = 0, ∀y ∈ X and satisfy the conditions of claim 3 above. Then we can assure that, for p → ∞,

L(t^v_pf_y)(y) = L(t^v_pf )(y) − Lf (y)

Lϕ^∗(y)L(t^v_pϕ^∗)(y) −→ 0,

uniformly in v and uniformly in y. Now the proof is over just observing that L(t^v_pϕ^∗)(y) → Lϕ^∗(y) uniformly in v and y.

3 Applications to C

([0, 1])

Now we state two results for sequences of operators defined on C^k([0, 1]). We omit the proofs since they are analogous to the ones of parallel results for the usual convergence that can be found in [6]. We shall denote ej(x) = x^j.

Let σ = {σ_i}_i≥0 be a sequence with σ_i ∈ {−1, 0, 1}. Let h, k ∈ N0 := N ∪ {0} with 0 ≤ h < k and σ_hσ_k 6= 0. We define the following cones in C^k([0, 1]):

C_h,k(σ) = {f ∈ C^k([0, 1]) : σ_iDⁱf ≥ 0, h ≤ i ≤ k}.

Let Γ = {i : h ≤ i < k, σ_i 6= 0, σ_i+1 = 0, σ_iσ_i+26= −1}. If Γ = ∅, then we call C_h,k(σ) a cone of type I, and if Γ 6= ∅, then we call C_h,k(σ) a cone of type II.

We denote σ^[j] = {σ^[j]_i }_i≥0 with σ_i^[j]= 0 for i 6= j and σ_j^[j]= σ_j.

Corollary 1 Let C_h,k(σ) be a cone of type I or II. Let K_n: C^k([0, 1]) → C^k([0, 1]) be a sequence of linear operators.

If K_n(C_h,k(σ)) ⊂ C_h,k(σ^[k]) and D^k(K_ne_j)−→ D^{a.c. k}e_j for every j = h, ..., k + 2, then D^k(K_nf )−→ D^{a.c. k}f ∀f ∈ C^k([0, 1]).

Corollary 2 Let C_h,k(σ) be a cone of type II and let r ∈ Γ. Let K_n : C^k([0, 1]) → C^k([0, 1]) be a sequence of linear operators.

If K_n(C_h,k(σ)) ⊂ C_h,k(σ^[r]) and D^r(K_ne_j)−→ D^{a.c. r}e_j for every j = h, h + 1, ..., k, then D^r(K_nf )−→ D^{a.c. r}f ∀f ∈ C^k([0, 1]).

4 Applications in C

In [2, Theorem 5] it is shown a Korovkin-type condition for the almost convergence of positive operators defined on C_2π. It is stated as follows:

Theorem 2 Let K_n be a sequence of positive linear operators defined on C_2π. The se- quence K_n(ψ) is almost convergent to ψ, uniformly on [0, 2π], for each ψ ∈ C_2π, if and only if K_n(e₀), K_n(sin(·)) and K_n(cos(·)) are almost convergent respectively to e₀, sin(·) and cos(·), uniformly on [0, 2π].

Now we present a generalization of this result considering operators non necessarily positive.

For i ∈ N0 ∪ {+∞} we define the linear operators L_i : C_2π → C_2π as follows: let φ ∈ C_2π and let a_i, b_i be the Fourier coefficients associated to φ, then L₀φ(x) = ^a₂⁰ and for i 6= 0,

L_iφ(x) = a₀ 2 +

i

X

k=1

a_kcos(kx) + b_ksin(kx).

Let h, k, m be integers with 0 ≤ h ≤ k < m. Let us consider the following cones in C_2π: C_2π^h,k = {φ ∈ C_2π: L_iφ ≥ 0, h ≤ i ≤ k}.

Corollary 3 Let P_m = {φ ∈ C_2π : L_mφ ≥ 0}. Let K_n : C_2π → C_2π be a sequence of linear operators.

If K_n(C_2π^h,k ∩ P_m) ⊂ P_m and L_m(K_nψ) −→ L^a.c. _mψ for all ψ ∈ {e₀, sin(i·), cos(i·) : i = 1, 2, . . . , k + 1}, then

L_m(K_nφ)−→ L^a.c. _mφ ∀φ ∈ C_2π.

Proof We apply Theorem 1 with A = B = C_2π, L = L_m, C = C_2π^h,k, where we consider C2π = C(X), being X identified with the compact set S = {z ∈ R²/|z| = 1}.

The condition (v.1) is trivially verified by u = e₀ ∈ C_2π.

Let z ∈ X, we define ϕ_z(x) = 1 + _2!¹ + _3!¹ + · · · + _(k+1)!¹ − cos(x − z) − _2!¹ cos²(x − z) − _3!¹ cos³(x − z) − · · · − _(k+1)!¹ cos^(k+1)(x − z) ∀x ∈ X. As the function ϕ_z admits a Fourier expansion with no null coefficients till order (k + 1) ≤ m, then for all x ∈ X, L_mϕ_z(x) = ϕ_z(x) ≥ 0, and the equality is satisfied if and only if x = z. Moreover, if h > 0, then Li(ϕz(x)) = 1 +_2!¹ +_3!¹ + · · · +_(k+1)!¹ − cos(x − z) −_2!¹ cos²(x − z) −_3!¹ cos³(x − z) − · · · − _i!¹ cosⁱ(x − z) ≥ 0, ∀x ∈ X, ∀i ∈ {h, h + 1, ..., k}, so ϕ_z ∈ C_h,k^2π, and if h = 0, then L₀ϕ_z(x) = 1 + 1

2! + 1

3!+ · · · + 1

(k + 1)! > 0 trivially. Hence, (v.2) is verified.

Now we show that condition (v.3) is also satisfied. Given φ ∈ C_2π, for i = h, h + 1, . . . , k, let m_i = min{(L_iφ)(x), x ∈ X}. Now if we consider β = max_h≤i≤k{(k + 1)!|m_i| + 1}, then for i = h, h + 1, ..., k

Li(βϕz+ φ)(x) = β(Liϕz)(x) + (Liφ)(x) ≥ |mi|(k + 1)!(Liϕz)(x) + (Liϕz)(x) + mi

≥ |m_i| + 1

(k + 1)!+ m_i ≥ 1

(k + 1)! > 0,

and therefore βϕ_z+ φ ∈ C_2π^h,k. The rest of the conditions are verified obviously.

Remark 2 Theorem 2 appears considering m = ∞ and k = 0.

References

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80, (1948), 167-190.

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